Elementary uses of Groebner bases

\def\P{\bf P} In this tutorial we introduce a number of basic operations using Groebner bases, and at the same time become familiar with a range of useful Macaulay2 constructs. The sections are:

A. First steps; example with a monomial curve

B. Random regular sequences

C. Division with remainder

D. Elimination theory

E. Quotients and saturation \beginsection{ A. First Steps; example with a monomial curve}\par To compute the Groebner basis of an ideal $(x^2y,xy^2+x^3)$ in the polynomial ring in four variables we proceed as follows:

Our favorite field

The Hilbert series of $M$ (the generating function for the dimensions of the graded pieces of $M$) is

Another way to get information about the module $M$ is to see its free resolution

In this section we shall first give a simple direct treatment of the problem above, and then show how to use Macaulay2's general tool to solve the problem. We first clear the earlier meaning of {\tt x} to make it into a subscripted variable

Having shown the primitive method, we now show a much more flexible and transparent one: we set up a ring map from the polynomial ring in $3$ variables (representing the plane) to $R/I$, taking the variables $y$ to the three linear forms that define the projection center. Then we just take the kernel of this map! (``Under the hood'', Macaulay2 is doing a more refined version of the same computation as before.) Here is the ring map

If $I$ and $J$ are ideals in a ring $R$, we define $(I:J)$, the ideal quotient, by $$(I:J) = \{f \in R \mid fJ \subset I\}.$$

In our first examples we consider the case where $J$ is generated by a single element $g$. This arises in practice, for example, in the problem of homogenizing an ideal. Suppose we consider the affine space curve parametrized by $t \mapsto (t,t^2,t^3)$. The ideal of polynomials vanishing on the curve is easily seen to be $(b-a^2, c-a^3)$ (where we have taken $a,b,c$ as the coordinates of affine space). To find the projective closure of the curve in $\P^3$, we must homogenize these equations with respect to a new variable d, getting $dc-a^2, d^2c-a^3$. But these forms do NOT define the projective closure! In general, homogenizing the generators of the ideal $I$ of an affine variety one gets an ideal $I_1$ that defines the projective closure UP TO a component supported on the hyperplane at infinity (the hyperplane $d=0$). To see the ideal of the closure we must remove any such components, for example by replacing $I_1$ by the union $I_2$ of all the ideals $(I_1:d^n)$, where $n$ ranges over positive integers. This is not so hard as it seems: First of all, we can successively compute the increasing sequence of ideals $(I_1:d), (I_1:d^2), \ldots $ until we get two that are the same; all succeeding ones will be equal, so we have found the union. A second method involves a special property of the reverse lex order, and is much more efficient in this case. We shall illustrate both. First we set up the example above:

A more perspicuous way of approaching the computation of the union of the $(I:d^n)$, which is called the saturation of $I$ with respect to $d$, and written $(I:d^\infty)$, is first to compute a reverse lex Groebner basis.